An elementary proof of the Brezis and Mironescu theorem on the boundedness and continuity of the composition operator: Ws,P(Rn) n W1,sp(Rn) ? Ws,p(Rn) is given. The proof includes the case p = 1.
Devoted to the memory of the applied mathematician Erhard Meister (1930-2001). This work is divided into two parts. Part A contains reminiscences about the life of E Meister. Part B displays the wide range of his scientific interests through eighteen papers with close scientific and personal relations to Erhard Meister
Classical boundary integral equations of the harmonic potential theory on Lipschitz surfaces are studied. We obtain higher fractional Sobolev regularity results for their solutions under sharp conditions on the surface. These results are derived from a theorem on the solvability of auxiliary boundary value problems for the Laplace equation in weighted Sobolev spaces.
The article is concerned with the Bourgain, Brezis and Mironescu theorem on the asymptotic behaviour of the norm of the Sobolev-type embedding operator: Ws,p ? Lpn/(n-sp) as s ? 1 and s ? n/p. Their result is extended to all values of s ? (0, 1) and is supplied with an elementary proof. The relation is proved. © 2002 Elsevier Science (USA).
We prove the Gagliardo-Nirenberg type inequality where 0 < ? < 1, 0 < s < 1, 1 < p < 8, and ?u? Ws,p is the seminorm in the fractional Sobolev space Ws,p (Rn). The dependence of the constant factor in the right-hand side on each of the parameters s, ?, and p is precise in a sense. © 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
We prove new point-wise inequalities involving the gradient of a function u is an element of C-1(R-n), the modulus of continuity w of the gradient delu, and a certain maximal function M(lozenge)u and show that these inequalities are sharp. A simple particular case corresponding to n = 1 and w(r) = r is the Landau type inequality [GRAPHIC] where the constant 8/3 is best possible and [GRAPHIC]
The goal of this work is to study the inhomogeneous Dirichlet problem for the Stokes system in a Lipschitz domain Omega aS dagger a"e (n) , na (c) 3/42. Our main result is that this problem is well posed in Besov-Triebel-Lizorkin spaces, provided that the unit normal nu to Omega has small mean oscillation.
We study the Dirichlet problem, in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order with bounded, complex-valued coefficients. A sharp corollary of our main solvability result is that the operator of this problem performs an isomorphism between weighted Sobolev spaces when its coefficients and the unit normal of the boundary belong to the space VMO.
A Brezis–Gallouet–Wainger logarithmic interpolation-embedding inequality is proved for various classes of irregular domains, in particular, for power cusps and λ-John domains.
Presentation of new results on the latest topics of the theory of Sobolev spaces, partial differential equations, analysis and mathematical physics
This volume is dedicated to the centenary of the outstanding mathematician of the XXth century Sergey Sobolev and, in a sense, to his celebrated work On a theorem of functional analysis published in 1938, exactly 70 years ago, where the original Sobolev inequality was proved. This double event is a good case to gather experts for presenting the latest results on the study of Sobolev inequalities which play a fundamental role in analysis, the theory of partial differential equations, mathematical physics, and differential geometry. In particular, the following topics are discussed: Sobolev type inequalities on manifolds and metric measure spaces, traces, inequalities with weights, unfamiliar settings of Sobolev type inequalities, Sobolev mappings between manifolds and vector spaces, properties of maximal functions in Sobolev spaces, the sharpness of constants in inequalities, etc. The volume opens with a nice survey reminiscence My Love Affair with the Sobolev Inequality by David R. Adams.
This is a survey of results mostly relating elliptic equations and systems of arbitrary even order with rough coefficients in Lipschitz graph domains. Asymptotic properties of solutions at a point of a Lipschitz boundary are also discussed.
The purpose of this book is to give a comprehensive exposition of the theory of pointwise multipliers acting in pairs of spaces of differentiable functions. The theory was essentially developed by the authors during the last thirty years and the present volume is mainly based on their results. Part I is devoted to the theory of multipliers and encloses the following topics: trace inequalities, analytic characterization of multipliers, relations between spaces of Sobolev multipliers and other function spaces, maximal subalgebras of multiplier spaces, traces and extensions of multipliers, essential norm and compactness of multipliers, and miscellaneous properties of multipliers. Part II concerns several applications of this theory: continuity and compactness of differential operators in pairs of Sobolev spaces, multipliers as solutions to linear and quasilinear elliptic equations, higher regularity in the single and double layer potential theory for Lipschitz domains, regularity of the boundary in $L_p$-theory of elliptic boundary value problems, and singular integral operators in Sobolev spaces.
Necessary and sufficient conditions for a function to be a multiplier mapping the Besov space B-1(m)(R-n) into the Besov space B-1(l)(R-n) with integer l and m, 0 < l <= m, are found. It is shown that multipliers between B-1(m)(R-n) and B-1(l)(R-n) form the space of traces of multipliers between the Sobolev classes W-1(m+1)(R-+(n+1)) and W-1(l+1)(R-+(n+1)).
An estimate of the Wiman-Valiron type for a maximum modulus on a polydisk of an entire function of several complex variables is obtained. The estimate contains a weight function involved also in the calculation of the radius of the admissible ball.
[No abstract available]